By investigating the evolution of the expectation value of operators,
we have demonstrated that MLP regression accurately extends
calculations in a highly reduced parameter space using very few
training examples. To understand the significance of this, a
comparison between the computational resources used in TEBD and MLP
calculations is presented. For TEBD calculations, long-time dynamics
are obtained by determining the state of the system
\(\mathinner {|{\Psi _{\mathrm {MPS}}}\rangle }\)  at every time step.[1]} For
\(N\)  sites with maximal bond dimension \(D\) , this results in using
\(O(poly(N)poly(D))\)  steps, more specifically, \(O(2w^3ND^3)\)  steps for
the sequential application of one- and two-body operators, where \(w\) 
is the matrix dimension of the applied local operator.[2]}
Computations with this complexity quickly become cumbersome for long
times, particularly when the correlation length of the system diverges
and the bond dimension \(D\)  scales exponentially. However, the MLP
regression circumvents this computational cost by utilizing a small
fixed "memory" of previously generated expectation values as the basis
for extending calculations out to long times. As can be seen in
Table REF , the computational cost of the combined training
and prediction steps of the regression is approximately polynomial. To
understand how this short "memory" reduces the complexity, consider a
training set having \(N_{\mathrm {train}}\)  examples constructed over
training windows (i.e., "memories") of size \(p\)  input into a neural
network have \(m\)  neurons. For a given value of \(p\)  elements, the
training phase of the MLP regression has a computational cost which is
determined entirely by the neural network model parameters, \(O(m^2p^2N_{\mathrm {train}})\) . After the training, prediction for later times
only has a computational cost of \(O(1)\) . By generating the first few
expectation values with the MPS, the MLP regression is shown to be
able to predict long-time operator expectations values with only the
addition of a relatively small number of compute cycles. We conclude
that within the regime of sizes considered in this study, the
computational cost of the MLP regression scales remarkably slowly
(nearly constant).
